Local linearity is a fundamental concept in calculus that describes how a function can be approximated by a line in a small neighborhood around a point. This concept is crucial for understanding derivatives and their applications.
Local Linear Approximation
Tangent Lines
A tangent line to a curve at point is a line that “touches” the curve at that point and has the same slope as the curve at that point.
Definition:
The tangent line to the graph of $ y=f(x) $ at the point $ (a,f(a)) $ is the line that passes through $ (a,f(a)) $ and has slope $ f’(a) $ , the Derivative of $ f $ at $ x=a $ .
Equation of the Tangent Line:
Using the point-slope form of a line, the equation of the tangent line is given by:
$$ y - f(a) = f’(a)(x-a) $$ Example:
Consider the function $ f(x) = x^2 $ . The Derivative of $ f(x) $ is $ f’(x) = 2x $ . At the point $ (1,1) $ , the slope of the tangent line is $ f’(1) = 2 $ . Therefore, the equation of the tangent line is: $$ y - 1 = 2(x-1) $$
y = x^2
y - 1 = 2(x-1)